C. Montella * Abstract. Keywords: Potential step; chronoamperometry; chronocoulometry; absorption; intercalation; diffusion coefficient.

Transcription

1 J. Electroanal. Chem., sous presse Discussion of the potential step metho for the etermination of the iffusion coefficients of guest species in host materials. I. Influence of charge transfer kinetics an ohmic potential rop C. Montella * École Nationale Supérieure 'Électrochimie et 'Électrométallurgie e Grenoble, Laboratoire Électrochimie et e Physicochimie es Matériaux et Interfaces, UMR 563 CNRS-INPG, associé à l UJF, Domaine Universitaire, B.P. 75, 384 Saint Martin 'Hères, France Abstract Theoretical expressions are given for the output response of ion-insertion electroes to a potential step assuming linear iffusion, restricte (blocking) iffusion conitions an possible limitations by insertion reaction kinetics. The effects of ohmic potential rop are also investigate. It is shown that slow interfacial charge transfer cannot be istinguishe from ohmic rop effects, in contrast to impeance iagrams where ohmic rop an charge transfer effects can be separate. The influence of potential step amplitue is iscusse. Chronocoulometric analysis is ealt with consiering iffusion controlle processes as well as mixe control conitions. The error in the etermination of the chemical iffusion coefficient of a guest species from chronoamperometric ata, when using the limiting Cottrell equation in the short-time range or the exponential ecay of current in the long-time omain, is evaluate in relation to insertion reaction kinetics an ohmic potential rop. Determination of the iffusion coefficients by curve fitting is also envisage using the current vs. time an charge vs. time relationships. Finally previous results in the electrochemical literature are iscusse in the light of the theoretical erivations propose in this paper. Keywors: Potential step; chronoamperometry; chronocoulometry; absorption; intercalation; iffusion coefficient. * Member of the Institut es Sciences et Techniques e Grenoble Tel.: ; fax: ; Claue.Montella@lepmi.inpg.fr

2 - -. INTRODUCTION The potential step chronoamperometry (PSCA) metho is wiely use for the etermination of the iffusion coefficients of guest species in host materials [] base on the theoretical erivation presente initially by Wen et al. [] for iffusion controlle processes. Moreover, the potentiostatic intermittent titration technique (PITT) [, 3], which is base on a staircase potential signal, is an extension of this metho. First, the iffusion moel, the ifferent graphical representations of the current transient use in the electrochemical literature to characterize insertion processes an the principles for the etermination of the iffusion coefficients of guest species from PSCA ata are reviewe. Next, some examples of eviation from the iffusion moel are quote an finally the aim of this paper is presente in the introuction... Diffusion moel A restricte (or blocking) iffusion conition is satisfie for a planar electroe, i.e. the iffusion flux is zero at some istance L from the electrolyte electroe interface, either consiering linear iffusion in a thin film of host material, of thickness L, eposite on a substrate impermeable to the iffusing species (Fig. A) or linear iffusion in a material foil or in platelet particles, of thickness L, symmetrically submitte to insertion on both sies (Fig. B). In the latter case, restricte iffusion is ue to the symmetry for the concentration profile with respect to the mile plane at the abscissa L. Assuming iffusion control (very fast insertion reaction kinetics), linear iffusion an restricte iffusion conitions, neglecting the effects of ohmic potential rop an ouble-layer charging current, finally isregaring phase transition processes, equivalent expressions of the output response to a potential step have been erive in the electrochemical literature. The following series which converges rapily in the long-time omain ( t > τ ) can for example be use [, 4, 5]: I Q π t ( t) = exp ( n ) τ 4 τ or alternatively: n = () Q n I( t) = + ( ) exp ( n t) t τ () πτ n = which is well suite to calculation for short times ( t < τ ). In the above equations, I is the

3 - 3 - iffusion current, t the elapse time from the beginning of the step, τ the iffusion time constant an Q the total amount of Faraaic charge passe following a potential step, with: τ = L D, Q = I () t t = FAL c (3) where L is the iffusion length, D the chemical iffusion coefficient of guest species, F the Faraay constant, A enotes the electrochemically active surface area an c the variation of guest species concentration in the host material ue to the applie potential step. Equilibrium is assume for the insertion reaction both at the initial an final times, an positive an negative values of c correspon to insertion an e-insertion processes, respectively. Minus sign in Eq. (3) applies to cation-insertion reactions ( H, Li, Na, K +, etc. ), so the Faraaic current an the relate charge are both negative (reuction reaction) for insertion processes ( E <, Q < an c > ) in accorance with the IUPAC convention. Neglecting all exponential terms in Eq. (), the well known Cottrell relationship is obtaine for short times (st): Q I () t st = ICottrell ()= t πτ t = FA D c πt (4) while an exponential ecay of iffusion current with respect to time is preicte for long times (lt), consiering the leaing term in Eq. (), as: I Q t t FA D π L c π Dt ( ) lt = exp = exp τ 4τ 4L (5).. Representations of the current transient From Eqs. () an (), the current is plotte vs. time in Fig. A. The current is ivie by the scale factor, Q FA D L c τ = ( ), an the time ivie by the iffusion time constant, τ, to obtain imensionless quantities in the figure for the sake of generality. The current variation shows a rapi ecay, as usually observe for iffusion controlle processes. The ratio of the iffusion current to the limiting Cottrell relationship in Eq. (4) is shown in Fig. B. The theoretical representation in Fig. B is the imensionless form of the Cottrell function plot, It () tvs. log t, use by Levi an Aurbach [6] to represent their experimental ata for lithium ion insertion into graphite electroes. Deviation from semi-infinite linear iffusion, i.e. eviation from a horizontal straight line in a It () tvs. log tplot, can be visualize easily by plotting such a

4 - 4 - graph. In aition, the usual Cottrell plot (current vs. time / ) is given in Fig. C in imensionless form an compare to the Cottrell straight line (ashe line) plotte from Eq. (4). Finally, the ecimal logarithm of current is plotte vs. time in Fig. D, using imensionless notation, an compare to the asymptotic straight line (ashe line) preicte from Eq. (5). Less than % eviation from the limiting Cottrell equation is observe in Fig. B for t τ < 9., i.e. t τ >. 3 in the Cottrell plot of Fig. C, while the linear variation in Fig. D is satisfie (less than % eviation from the limiting current in Eq. (5)) when t τ > 3.. Hence, the transition between the two limiting expressions for the iffusion current, respectively semi-infinite linear iffusion current in Eq. (4) an exponential ecay in Eq. (5), is observe within a narrow time range near t =. τ for iffusion controlle processes an restricte linear iffusion conitions. Some authors note that the change in shape of the current transient is of greater importance than the absolute current level, an therefore it is very effective to introuce the erivative of the current transient or logarithmic current with respect to time or logarithmic time [7, 8]. The logarithm of iffusion current is plotte in Fig. 3B vs. logarithm of time, using imensionless notation. The transient ue to the iffusion process shows a linear epenence, log current vs. logtime, with a slope of ( /) characteristic of semi-infinite linear iffusion, followe by a steep exponential ecay ue to restricte iffusion. The erivative of logarithmic current with respect to logarithmic time is plotte in Fig. 3C with the constant value ( /) in the short-time omain. Such plots are use in the electrochemical literature to iscriminate between iffusion in single-phase materials, as stuie in this paper, an iffusion couple with phase transition processes in two-phase materials, such as lithium-ion insertion into Li electroes [7, 8]. CoO δ Finally, the graphical representation of Fig. 3D has not yet been use in electrochemical literature, to the best of our knowlege. Nevertheless, it is well suite to the characterization of iffusion controlle processes an restricte linear iffusion conitions because of the two limiting expressions respectively vali for short an long times: log I ( t) log t = / (6) st an:

5 - 5 - lt τ log I ( t) logt π t = ( ) 4 (7) The valiity omains for the above expressions are separate by a narrow time range near t =. τ, which is characteristic of restricte linear iffusion..3. Determination of the iffusion coefficient of a guest species The iffusion coefficient of a guest species is generally etermine either from the short-time expression of the iffusion current in Eq. (4) where ICottrell( t) t is the characteristic time-invariant function [6] an also the slope ( Sl I,st ) of the Cottrell straight line in a current vs. time / plot, from which D is obtaine as: () ICottrell t t I D L Q Q L I = = Sl,st = Sl,st π π π FA c (8) provie the iffusion length L is known inepenently, or from the long-time omain where an exponential ecay of current is preicte. Taking the ecimal logarithm of both sies in Eq. (5) an plotting log current vs. time, the iffusion coefficient can be obtaine from both the intercept ( Int I,lt an: ) an slope ( Sl I,lt ) of the portion of straight line, accoring to: L Int L I D = = Q FA c,lt Int I,lt (9) L ln D = 4 π Sl,lt () I More generally, using a curve fitting proceure, experimental chronoamperograms can be compare to Eqs. () an () over a large time range provie ohmic potential rop is isregare an the ouble-layer charging current is negligible compare to the Faraaic current. The total Faraaic charge Q passe following a potential step is obtaine by integration of the current transient with respect to time, so only one ajustable parameter, τ = L experimental chronoamperograms to Eqs. () an (). D, is neee to compare.4. Deviation from the iffusion moel Nevertheless, eviations from the theoretical relationships in Eqs. () to () an relate graphs in Figs. an 3 are frequently observe in experimental works focuse on host materials an

6 - 6 - insertion reactions. Recent examples inclue stuies on lithium-ion insertion in thin graphite, LixNiO an LixCo.Ni.8O electroes by Levi et al. [4, 6, 9] an Li-δ CoO electroes by Shin an Pyun [7, 8], potassium-ion insertion in thin films of prussian blue (ferric ferrocyanie, PB) by Garcia-Jareño et al. [], as well as other works not iscusse in this paper. Potassium-ion insertion in thin PB films eposite on ITO substrate was thoroughly examine by Garcia-Jareño et al. using PSCA [], cyclic voltammetry [] an the EIS metho []. Consiering chronoamperometric results, eviation from the Cottrell equation was analyze by the above authors in terms of ohmic potential rop in the electrolytic solution an the ITO substrate. Numerical simulations were carrie out by these authors, taking ohmic rop into consieration, an the theoretical moel was consistent with experimental ata. However, the impeance iagrams plotte for the same materials [] clearly show the presence of a large interfacial charge transfer resistance compare to the solution an substrate resistances, which enotes slow ion-transfer kinetics at the solution PB interface an/or slow electron-transfer kinetics at the PB ITO interface. Hence, the question to be answere is whether or not kinetic limitations by interfacial charge transfer can be neglecte when analyzing the current transient ue to a potential step for potassium-ion insertion in PB ITO electroes. On the other han, Levi et al. [4, 6, 9] stuie lithium-ion insertion in thin graphite, LixNiO an LixCo.Ni.8O electroes using cyclic voltammetry, chronoamperometry (PITT), current pulses an the EIS metho. Consiering potential step experiments, these authors always observe that the Cottrell region (Eq. (4)) is narrow with respect to time given that the short-time omain is affecte by ohmic potential rop, lithium-ion migration through superficial passivation layers on particles of host material an slow ion-transfer at the particle surface, while the long-time eviation from Eq. (5) is connecte with particle size istribution. Similarly, eviation from the restricte (or finite space) iffusion impeance was observe by these authors in the low frequency omain, consiering the impeance iagrams plotte for the same materials [6, 9]. Finally, Shin an Pyun [7, 8] observe that the log current vs. log time plot for lithium-ion insertion in Li-δ CoO electroes an potential stepping in the single α phase omain eviates from the theoretical preiction in Fig. 3B an shows a monotonic increase in its slope from an almost flat value to the infinite one in absolute value. These authors interprete their chronoamperometric ata in terms of purely cell-impeance controlle iffusion of lithium-ion in Li δ CoO [8]. -

7 This work The aim of this work is to point out an iscuss theoretically the effects of insertion reaction kinetics an ohmic potential rop on the output response of ion-insertion electroes to a potential step an to emonstrate the avantage of the simultaneous application of potential step an EIS methos for the etermination of the chemical iffusion coefficients of guest species in host materials. The theoretical response to a potential step epens on the geometric moel use to represent insertion an iffusion processes in host materials. In this paper we are concerne with planar films of host material, of thickness L, eposite on a substrate impermeable to the iffusing species (Fig. A), as generally assume for thin films obtaine by electroeposition, sputtering or spray eposition. Diffusion of guest species starts at the electrolyte film interface, while electron transfer is observe simultaneously at the substrate film interface. Ieal thin films (isotropic material, no structural efects) are assume, so that the iffusion coefficient of intercalate species takes on a simple meaning in this case ue to one-imensional iffusion. The moel use below for insertion an iffusion processes also applies to linear iffusion in material foils, of thickness L, symmetrically submitte to insertion on both faces (Fig. B). Levi an Aurbach [6] use the same geometric moel to represent lithium-ion insertion into platelet particles of graphite in thin film composite electroes. Two-phase interfaces are consiere in the above moel. A reaction front avancing from the three-phase junction, electrolyte substrate active material, was envisage alternatively by Schröer et al. [3] an Lovric an Scholz [4, 5] for moeling the propagation of a reox reaction through microcrystals (mixe ionic-electronic conuctor) immobilize at an electroe surface an in contact with an electrolyte solution. The influence of crystal shape an crystal size on the output response to a potential step was thoroughly examine in these papers using algebraic as well as numerical methos. A three-phase junction, electrolyte active particle electronic conuctor, may also be consiere in composite electroes where aitives, such as carbon black, are generally use to improve electronic conuction in the electroe, which results in a complicate geometry for couple masstransport of ions an electrons in electroactive particles. Two- or three-imensional iffusion occurring from a three-phase junction is isregare in this paper because of the simple geometric moel assume for the thin film electroes an particles

8 - 8 - in Fig.. Moreover, to focus on insertion reaction kinetics an ohmic rop effects, linear iffusion of guest species in single phase materials an/or potential omains (no phase transformation) is assume below. Other effects relate to the geometry of iffusion in host materials an the istribution of iffusion lengths in particles of composite electroes will be ealt with in the secon an thir parts of this work [6]. First the theoretical response to a potential step is erive in this paper an the possible shapes for the Faraaic current transient iscusse, next chronocoulometric analysis is ealt with an finally the conitions for the etermination of the iffusion coefficients of guest species are analyze. Previous results in the electrochemical literature are also iscusse in the light of the theoretical erivations given in this paper.. CHRONOAMPEROMETRY To our knowlege, the first theoretical work focusing on the response of insertion electroes to a potential step an taking into account kinetic limitations by surface processes is that of Krapvinyi et al. [7, 8] who stuie hyrogen extraction from metallic sheets using large-amplitue potential steps. More recently, Chen et al. [9] given the theoretical response to a small-amplitue potential step for hyrogen absorption in thin metal films or foils, uner restricte linear iffusion conitions, assuming no hyrie formation an taking kinetic limitations at the metal electrolyte interface into consieration. Using the metallic-type potential istribution in host material electrolyte systems, as iscusse by Levi an Aurbach [6], the moel in Ref. [9] also applies to ion-insertion processes in thin films, foils or platelet particles of host material with planar geometry. This moel is extene below to cover the case where ohmic potential rop is present. The insertion material is assume to be initially ( t < ) in the equilibrium state corresponing to an electroe potential E an a uniform concentration of guest species, c = c( E), given by the insertion isotherm c(e), which is the reverse function of the coulometric titration curve Ec () generally use in the literature. A potential step from E to E = E + E, where E can be either positive or negative, which correspons respectively to e-insertion an insertion processes for cations, is impose on the electroe at t =. After the current transient, the electroe tens towars the new equilibrium state corresponing to potential E an uniform concentration of guest species c = c( E ), with c = c + c.

9 Laplace transform of Faraaic current Assuming small-signal (linearity) conitions for both potential step an EIS methos, the Laplace transform of the current transient ue to a potential step from E to E E RT nf + E, with << ( ) where the number (n) of electron involve in the reaction is equal to one in this () paper an the other symbols have their usual meaning, is relate to the electroe impeance Zs measure at the steay-state potential E through the following equation where s enotes the Laplace complex variable: ()= Is E sz s () () Using the usual equivalent circuit for the impeance of electrochemical systems [], Zs () can be written as: [ ] ()= + + () Zs RΩ scl ZF s () where ZF( s) is the Faraaic impeance, C l the ifferential (ouble-layer) capacitance an R Ω the sum of ohmic resistances in the electrolyte, bulk material an material substrate or current collector. More complicate equivalent circuits are sometimes propose for insertion processes to inclue the effects of ion migration through superficial passivation layers when consiering lithium-ion insertion into graphite electroes [4, 9] for example. Provie the ifferential (ouble-layer) capacitance, as well as the possible capacitances of superficial layers on the host material (e.g. passivation layers on graphite particles), are sufficiently low with respect to the insertion capacitance of the bulk material, the electroe impeance can be approximate, except in the high frequency (or s) omain, by: Zs () RΩ + ZF () s (3) Assuming a uniformly accessible electroe, the Faraaic impeance for insertion reactions is well known. Neglecting limitations by mass transport in the electrolyte, consiering a irect (onestep) insertion reaction, assuming linear iffusion of guest species in a thin film or foil of host material or in platelet particles of a composite electroe (in the latter case the potential graient in the electroe is neglecte), with restricte iffusion conitions (Fig. ), finally isregaring phase transition processes, ZF( s) takes on the following expression [9,, ]: ZF()= s Rct + Rcoth τs τ s, τ = L D (4)

10 - - where the interfacial charge transfer resistance, R ct, an the iffusion resistance, R, are written in terms of thermoynamic, kinetic an iffusion parameters [9] as: R ct = ( FAv E ) (5) [ ] R FAm c E, m = D L (6) = ( ) In these equations, m is the iffusion constant, τ the iffusion time constant an c insertion isotherm slope which satisfies: E the c E = v v E c (7) because of equilibrium state at the initial potential of potential step. v E an v c are the partial erivatives of insertion reaction rate (v) with respect to potential an concentration of guest species, both evaluate at the initial potential E : v = v E E ( ), v = ( v c) (8) c c E The quantities c E, v E an v c are all negative in Eqs. (5) to (8) ue to the assumption of cation-insertion reactions in this paper. Moreover, consiering small signals (linearity conition), the concentration variation in the host material is relate to the potential step amplitue by the linearise relationship: c c E E (9) = ( ) Introucing the imensionless parameter: Λ = R R + R vc m = = + R R m v + R R Ω ct Ω ct c Ω () where all quantities are evaluate at the initial potential E, the Laplace transform of Faraaic current is erive from the above equations as: ()= Is ( ) E RΩ + Rct s + Λ coth τ s τ s ( ) Setting s in Eq. (), the term coth τ s τ s tens towars zero, an, using the inverse Laplace transform, the Faraaic current at the initial time is given by the equivalent expressions: ()

11 - - E E Q Ι( )= = Λ = Λ = ΛFA D R + R R τ L c Ω ct () The initial current epens linearly on the concentration variation in the host material, c, the total charge passe following a potential step, Q, an the potential step amplitue E. Hence, the Faraaic current vs. potential relationship follows Ohm s law for short times (t ) provie E << RT F. From Eqs. () an (), we obtain finally: ()= Is I ( ) ( ) s + Λ coth τ s τ s.. General expressions for the Faraaic current (3) Inverse Laplace transform of Eq. (3) gives the Faraaic current transient cause by a smallamplitue potential step as the infinite series expansion: Λ It ( ) = I( ) exp bn t τ (4) Λ Λ+ b n = + ( ) where b n is the n th positive root of the equation: n btan b Λ = (5) Equivalent formulations of It () are easily erive from Eqs. () an (4) as: an: E Λ It ( ) = exp bn t τ (6) R R Λ Λ+ b Ω + + ( ) ct n = n E Λ It ( ) = exp bn t τ (7) R Λ Λ+ b n = + ( ) n Eqs. (6) an (7) give the current transient in terms of electrical components of the equivalent circuit for the electroe impeance. These equations can also be written in terms of thermoynamic, kinetic, iffusion an ohmic parameters, from Eq. (), as: It FA D L c Λ ( ) = exp b Dt n + + b Λ Λ L n = n (8)

12 where minus sign applies to cation-insertion reactions ( H, Li, Na, K +, etc. ). A in Eq. (8) is the total surface area (both faces) when using the electroe or particle geometry of Fig. B, while it is the area of the material electrolyte interface (single face) for the thin film in Fig. A. Finally, using the iffusion time constant an the total amount of Faraaic charge passe following a potential step, as efine in Eq. (3), an alternative formulation of the current transient is erive from Eq. (8) as: Q Λ It ( ) = exp bn t τ τ Λ Λ b n = ( ) + + n The above relationship is vali for both electroe geometries in Fig. an generalizes the well known Eq. () which is obtaine as a limiting case setting Λ (iffusion control) in Eq. (9) with bn = ( n ) π from Eq. (5). Eq. (4) an Eqs. (6) to (9) give five equivalent formulations of the electroe response to a small-amplitue potential step for cation-insertion processes stuie by the PSCA metho uner restricte linear iffusion conitions, in the absence of phase transition processes. Due to the use of a small signal amplitue, the theoretical erivation is vali whatever the insertion isotherm type (Langmuir, Frumkin or other isotherm) an reaction rate expression (Butler-Volmer or other relationship). (9) The relationship between the chemical iffusion coefficient of a guest species an the insertion isotherm was erive first by Arman [3], to the best of our knowlege, an iscusse more recently by Chen et al. [4] an Levi an Aurbach [6]. However, Eqs. (4) to (9) apply whether the iffusion coefficient is constant or epens on the guest species concentration in the host material because of (i) a small signal amplitue, (ii) restricte iffusion conitions an therefore (iii) uniform concentration of guest species in the host material an equilibrium for the insertion reaction uner steay-state conitions [5]. The theoretical expressions for the current transient cause by a small-amplitue potential step have been erive in this Section assuming that the capacitive current is negligible compare to the Faraaic current. Therefore, these equations cannot be use to fit experimental ata obtaine for very short times where measurements are corrupte ue to the presence of the ouble-layer capacitance. In particular, the initial current in Eq. () cannot be measure irectly an extrapolation techniques shoul be use to etermine I( )..3. Parameter Λ: a key factor for the kinetics of insertion processes

13 - 3 - Clearly, the imensionless parameter Λ is a key factor for the kinetics of insertion processes stuie by PSCA. In Eq. (), vc m compares the iffusion constant, m = D L, an the reaction rate constant which is inclue in the partial erivative v c. In aition, the influence of ohmic potential rop is characterize by the resistance ratio RΩ Rct or R R Ω. The general expression for the parameter Λ, irrespective of insertion isotherm type an rate control conitions, is obtaine from Eqs. (5) an (), in terms of partial erivatives of reaction rate, as: Λ = vc L D + RFAv Ω E while equivalent formulations can be written using the insertion isotherm slope, c erivative calculus rule in Eq. (7). (3) E, an the The Λ value for PSCA experiments epens on the insertion isotherm type, the initial potential of potential step, the reaction rate constant, the iffusion coefficient of guest species, the film, foil or particle thickness in Figs. A an B, an the presence of ohmic rop. For example, assuming Langmuir isotherm conitions for the insertion process (formation of a soli solution with no interaction in the host material) an a cation concentration of mol L (stanar concentration) in the electrolyte solution, Λ takes on the following expression erive in the Appenix A: ( ) ( )( + ) ( ) ( ) k L D exp αξ r expξ Λ = + R ffak c exp α ξ expξ Ω max [ r ] + where k is the stanar rate constant for the insertion reaction, α r the symmetry factor of interfacial charge transfer in the irection of reuction, c max the maximal (saturation) concentration of guest species in the host material, ξ the initial potential of potential step written in imensionless form, ξ = f( E E ) with f = F ( RT), an E enotes the stanar potential, i.e. the equilibrium potential corresponing to the stanar concentration in the soli solution, c = c max. (3) On the other han, whatever the insertion isotherm, Λ can be estimate from Eq. () using experimental values of the iffusion resistance, charge transfer resistance an electrolyte an/or substrate resistances measure on an impeance iagram plotte for the same potential range.

14 Parameter Λ an concentration profile in the host material The influence of the parameter Λ is well illustrate by the concentration profile vs. time variation plotte in Figs. 4 to 6. Solving the iffusion equations together with the bounary conitions at the material surfaces (continuity conition at x = an blocking iffusion conition at x = L), the Laplace transform of guest species concentration in the host material is given by: ( ) [ ( )] [ ] c I cosh τ x L s cxs (, )= s FAms Λ cosh τ s + τ s sinh τ s, x L (3) where x is the istance from the electrolyte host material interface. Using inverse Laplace transforms of the ifferent terms, we obtain, after some reorganization, the normalize concentration vs. space an time epenence ue to potential stepping: cxt (, ) c Λ = c c Λ + Λ+ b n = ( n ) [ ( )] cos bn x L cosb n ( n ) exp b t τ (33) This relationship can alternatively be obtaine from the theoretical expression given by Crank [6] for similar problems of iffusion in a plane sheet couple with surface evaporation processes, because of the same iffusion equations an bounary conitions once written in imensionless form. Large Λ values are characteristic of iffusion controlle processes ue to very fast insertion reaction kinetics an negligible ohmic rop or IR compensation. A concentration step from c to c is impose at the electrolyte host material interface ( x = ) when Λ, ue to the applie potential step from E to E, an the mass transport process varies from semi-infinite iffusion conitions for short times to restricte iffusion conitions for long times (Fig. 4). The relate limiting expression for the concentration is erive, setting Λ in Eqs. (5) an (33), as: ( ) cxt, Λ c 4 x t = sin π ( n ) π exp ( n ) c c ( n ) L (34) π 4 τ n = In opposition, consiering small Λ values ue to slow charge transfer kinetics at the interface an/or large ohmic rop in the ifferent phases, we have b = Λ an b nπ (for n > ) from n = Eq. (5), so all exponential terms, except the first, can be neglecte in Eq. (33) when Λ tens towars zero. It follows that: ( ) cxt, c c c Λ = exp( Λ t τ ) (35) c epens only on the time variable uner the above conitions an quasi-uniform concentration of

15 - 5 - guest species (negligible concentration graient in the host material) is preicte over the whole time omain, as shown in Fig. 5. Finally, intermeiate values of the parameter Λ in Eqs. () an (3) are characteristic of mixe control conitions, i.e. control by iffusion an interfacial charge transfer kinetics an/or ohmic rop. An example for such a concentration profile vs. time variation is presente in Fig. 6. The mass transport process varies from semi-infinite iffusion conitions for short times to restricte iffusion conitions for long times, but, in contrast to Fig. 4, the interfacial concentration of guest species is time epenent, which is generally the case for most real systems..5. Influence of capacitive current an ohmic rop In Ref. [9] focuse on hyrogen absorption in thin metal films, we wrote: We have not taken into consieration the electrolytic solution resistance R s in the preceing calculation as the influence of such a resistance an of the ifferential (ouble-layer) capacitance C l can be neglecte for times longer than the time constant, τ = RR R + R C, whose value.... [ ( )] s ct s ct l This is incorrect an shoul be replace by: uner restricte iffusion conitions, the current transient cause by a small potential step is not moifie, ue to the presence of the ifferential (ouble-layer) capacitance, for times longer than the time constant, τ = [ RR s ct ( Rs + Rct )] Cl, provie C l takes on a very low value with respect to the insertion capacitance of the host material, i.e. Cl << FAL c E, which is generally observe for most real systems. However, if the latter conition is not satisfie, the ouble-layer capacitance influences the response to a potential step also in the long-time range ue to the blocking conition, JLt (, )=, for the iffusion process in the host material. On the other han, whatever the kinetic an iffusion conitions, uncompensate ohmic rop affects the current transient over the whole time omain, in opposition to the assertion in Ref. [9]..6. Influence of insertion reaction kinetics an ohmic rop The shape of the current vs. time curve calculate from Eqs. (5) an (9) epens only on the imensionless parameter Λ given that the ratio Q τ an the iffusion time constant τ are scale factors for current an time, respectively. Looking at Eq. (), we note that the effects of slow iontransfer kinetics at the electrolyte host material interface, as well as slow electron-transfer kinetics at the host material substrate (or current collector) interface, cannot be istinguishe from ohmic rop effects. Due to slow interfacial charge transfer (high R ct value compare to R ) an/or large ohmic rop (high R Ω value), the Λ value ecreases in Eq. () an the current transient ue to a

16 - 6 - potential step is affecte over the whole time omain, as inicate by Eqs. (5) an (9). Hence, the effects of surface reaction kinetics an ohmic potential rop shoul be iscusse at the same time for the PSCA metho. In contrast, ohmic rop effects an kinetic limitations by interfacial charge transfer can be iscriminate using the EIS metho an the Nyquist representation of impeance iagrams (see Fig. 3B below). Hereafter, we assume that the ouble-layer charging current is negligible compare to the Faraaic current over the time range consiere an we use Eqs. (), (5) an (9) to moel the current transient. The current is plotte in Fig. 7 using the same imensionless representations as in Fig.. Large Λ values ( in the figure) give the same theoretical epenences as in Fig. ue to iffusion control. The limiting response observe for short times (st) epens on Λ. Diffusion control an semiinfinite linear iffusion (which correspons to the concentration profile of Fig. 4 for short times), i.e. the Cottrell equation, are satisfie when Λ an t, as shown in Figs. 7B an C: It I t Q πτ t () st, Cottrell Λ = ()= = FA D c πt (36) In contrast, interfacial charge transfer an/or ohmic rop control ( Λ an therefore b = Λ from Eq. (5)), which correspons to quasi-uniform insertion conitions (Fig. 5), yiels the exponential epenence over the whole time omain: Q E FA vc c v t L It ( ) Λ = Λ exp ( Λt )= c τ exp( Λt τ )= exp τ RΩ + Rct + RΩFAvE + RΩFAv (37) E which can be linearise for short times as: Q E FA vc c v t L It () = c st, Λ Λ ( Λt )= τ ( Λt τ)= τ RΩ + Rct + RFAv Ω E + RFAv Ω E (38) In the above equations, the partial erivative ( v c < ) is replace by ( v c ), so minus sign is apparent for cation insertion processes (reuction reaction, with E <, Q < an c > ). Finally, intermeiate Λ values, which correspon to mixe control by iffusion an insertion

17 reaction kinetics an/or ohmic rop, lea for short times (semi-infinite linear iffusion in Fig. 6 for t << τ ) to the relationships: Q E It ( ) st = Λ exp( Λ t τ) erfc( Λ t τ)= exp Λ t τ erfc Λ t τ τ RΩ Rct FA v c c vtd = c vc t D exp + erfc (39) RΩ FAvE ( + R FAv ) + R FAv Ω E Ω E which can be simplifie for very short times (vst), using Maclaurin series expansions of exponential function an error function complement, as: [ ] = ( ) ( ) [ ] E It () vst = Λ( Q τ) Λ t( πτ) Λ t ( πτ ) RΩ Rct FA v ( ) c c vc t π D = (4) + RΩFAvE + RΩFAvE In contrast, Eq. (39) becomes equivalent to (less than % eviation from) the Cottrell relationship in Eq. (36) when Λ t τ > Setting R Ω = (no ohmic rop or IR compensation) an H = Λ τ = vc D, Eqs. (39) an (4) are similar in shape to the relationships given by Bar an Faulkner [7] for one-step reox reactions with semi-infinite linear iffusion in the electrolyte an kinetic limitations at the electrolyte electroe interface. Nevertheless, the influence of ohmic rop is taken into consieration in Eqs. (39) an (4), in contrast to the erivation by the above authors. Deviation from iffusion controlle processes, ue to limitations by insertion reaction kinetics an/or ohmic potential rop, is well characterize by the presence of the maximum of the function in Fig. 7B, which is equivalent to a maximum of the Cottrell function plot, It () tvs. log t, from the experimental point of view. No portion of straight line is present in the Cottrell plot of Fig. 7C uner the above conitions. This is observe, for example, consiering the experimental results obtaine by Levi et al. [6, 9] for lithium-ion insertion into thin graphite electroes. The asymptotic behavior observe in the long-time omain is illustrate in Fig. 7D. For long times (large t τ ), restricte iffusion is satisfie in Figs. 4 to 6 whatever the Λ value. Only the first term of the series in Eq. (9) an the equivalent expressions of Faraaic current is consiere, so the absolute value of current shows an exponential ecay vs. time. It follows that a log It () vs. time plot shoul be linear for long times whether iffusion control ( Λ an b = π ): Q t E t It FA D R L c Dt ( ) = π π π lt, Λ exp = exp = exp τ 4τ 4τ 4L (4)

18 - 8 - or control by interfacial charge transfer an/or ohmic rop ( Λ an b = Λ): Q E FA vc c vc t L It ( ) lt, Λ = Λ exp ( Λt )= τ exp( Λt τ )= exp τ RΩ + Rct + RΩFAvE + RΩFAv (4) E or mixe control by iffusion an interfacial charge transfer an/or ohmic rop (intermeiate Λ value an b given by the first positive root of Eq. (5)) is satisfie: Q bt It FA D Λ ( ) exp b L c exp b Dt Λ lt = = b τ Λ Λ τ Λ Λ L ( ) ( ) (43) Levi et al. [6, 9] note that the short-time response of ion-insertion electroes to a potential step is moifie ue to slow ion-transfer kinetics at the host material surface. More generally, slow interfacial charge transfer an ohmic potential rop affect the current transient over the whole time omain as shown in Fig. 7. Moreover, an important feature of Eqs. (4) to (43) is that an exponential ecrease of current with respect to time, in the long-time range, is not a sufficient conition for preicting a iffusion controlle rate for insertion processes in thin films, foils or particles of host material. The logarithm of current is plotte vs. logarithm of time in Fig. 8B, using the same imensionless representation as in Fig. 3B, an the erivative of this curve is given in Fig. 8C. The influence of insertion reaction kinetics an ohmic rop is clearly shown in the short-time omain where log It ( ) logt varies from ( /) for iffusion controlle processes to zero ue to constant current at short times for interfacial charge transfer an/or ohmic rop control. The effects of insertion reaction kinetics an ohmic rop on the current transient are also illustrate in Fig. 8D. Diffusion control ( Λ ) gives the same characteristic curve as in Fig. 3D with the limiting value ( /) for short times an the limiting slope, π ( τ ) 4, for long times in a log It ( ) logt vs. time plot. In contrast, log It ( ) logt tens towars zero when t ue to control by insertion reaction kinetics an/or ohmic rop, while a portion of straight line is preicte in the long-time range, whatever the Λ value, accoring to: log It ( ) logt bt lt = τ (44)

19 - 9 - Hence, eviation from iffusion controlle processes, ue to limitations by interfacial charge transfer kinetics an/or ohmic rop, is well characterize by the value ( log It ( ) logt> ) note in the short-time omain. This is observe, for example, in the papers by Shin an Pyun [7, 8] for lithium-ion insertion in Li-δ CoO electroes an potential stepping in the single α phase omain. The experimental results in Ref. [8] were interprete by these authors assuming purely cell-impeance controlle iffusion of lithium-ion in Li-δ CoO, i.e. assuming that the currentpotential relationship is purely ohmic. This moel is an alternative to the theoretical erivation presente in this work. The two moels will be compare in a separate paper. As a first conclusion, it shoul be emphasize that the use of the classical iffusion laws in thin film materials, i.e. Eqs. () an () above, is only vali when consiering very fast insertion reaction kinetics, negligible ohmic rop or IR compensation, an therefore iffusion controlle processes. In contrast, Eqs. (4), (8) an (9) erive in this paper apply for kinetic, iffusion an ohmic rop control, as well as mixe control conitions. In aition, the current transient can be moele alternatively in terms of electrical components of the equivalent circuit for the electroe impeance in Eqs. (6) an (7). Whatever the expression consiere for the current transient, the key factor for the kinetics of insertion processes stuie by the PSCA metho is the imensionless parameter Λ in Eqs. () an (3).7. Potential step metho vs. EIS metho As inicate above, the Laplace transform of the current transient cause by a potential step an the electroe impeance measure at the same initial potential are relate through Eq. () provie a sufficiently small potential step is consiere. Uner the above conitions irect corresponences can be state between the results obtaine by the two methos. The limiting Cottrell behavior observe in the short-time omain correspons to the Warburg impeance satisfie for high frequencies, the exponential ecay of current note for long-time chronoamperometric experiments is relate to the low frequency capacitive behavior in EIS ata, etc. Hence, the same theoretical moel shoul be use to fit experimental chronoamperograms an EIS ata collecte for the same material. Corresponences between PSCA an EIS ata were note in the experimental work by Levi an Aurbach [6] focuse on lithium-ion insertion in thin graphite electroes. These authors note that chronoamperometric ata eviate from Eq. () in the long-time range an EIS ata are not fitte well by the restricte iffusion impeance in the low frequency omain, ue to particle size istribution. Levi an Aurbach use the so-calle Frumkin an Melik-Gaykazyan (FMG) impeance to fit their EIS ata. However, the FMG impeance is given by the series combination of a boune iffusion impeance [8] (or finite length iffusion impeance in Ref. [6]) an the

20 - - insertion capacitance; in contrast, restricte iffusion (finite space iffusion accoring to Ref. [6]) is generally assume for insertion materials. Moreover, these authors propose no theoretical expression for the output response of a composite electroe to a potential step that is consistent with the FMG impeance. Therefore, the use of the FMG impeance cannot be recommene to fit experimental ata collecte for ion-insertion materials..8. Influence of potential step amplitue A small-amplitue potential step was assume in the above Sections of this paper in orer to linearise the kinetic equations. The situation is more complicate when consiering non-linear conitions ue to a large potential step amplitue, as use experimentally by Garcia-Jareño et al. [] for stuying potassium-ion insertion in PB ITO electroes. These authors impose a potential step from the initial potential, E =.6 V vs. Ag/AgCl/KCl M reference electroe, corresponing to the prussian blue omain, to the final potential, E =. V vs. the same reference, which correspons to the prussian white (or Everitt salt) omain. The same remark applies to the experiments of Shin an Pyun [7, 8] focusing on lithium-ion insertion in Li electroes. These authors impose potential steps with amplitues of several hunre of mv. CoO -δ No general analytical solution of the iffusion an kinetic equations is possible uner the above conitions an numerical calculation, using a finite ifference metho for example, is neee to simulate the theoretical behavior of the insertion electroe. However, a particular situation can be treate theoretically in close form to unerstan the influence of potential step amplitue. Assuming that the iffusion coefficient of guest species oes not epen on its concentration insie the host material, an therefore on the electroe potential, that Langmuir isotherm conitions (formation of an ieal soli solution with no interaction in the host material) are satisfie for the insertion process, an isregaring ohmic rop an ouble-layer charging current effects, the theoretical erivation presente for small-amplitue potential steps in Eqs. (4), (8) an (9) above can be generalize to large potential steps [9] provie (i) the expression for the imensionless parameter Λ in Eq. () is replace by: Λ = v c m (45) because of the assumption of a negligible ohmic rop or IR compensation, an (ii) the partial erivative of reaction rate with respect to guest species concentration is evaluate at the applie potential E rather than the initial potential E. Uner the above conitions, the parameter Λ takes on the expression:

21 - - ( ) ( αξ) ( + expξ ) Λ = k L D exp, ξ = f E E (46) r ( ) an Eqs. (4), (8) an (9) remain vali for large potential step. In contrast, the linear epenence of the concentration variation c on the potential step amplitue E in Eq. (9) is not satisfie ue to a large signal amplitue an therefore Eqs. (6) an (7) cannot be use to fit experimental ata for large potential steps. Disregaring ohmic rop effects, large Λ values (iffusion control) can be achieve for thin films, foils or platelet particles of host materials, an therefore the limiting relationships in Eqs. () an () satisfie, either consiering a small-amplitue potential step an a very high insertion rate constant, irrespective of the insertion isotherm type an the epenence or not of the iffusion coefficient on the electroe potential, or using a sufficiently large potential step amplitue, irrespective of insertion reaction kinetics, provie D is constant. In contrast, in the presence of ohmic rop, Garcia-Jareño et al. [] neglecte kinetic limitations by interfacial charge transfer for potassium-ion insertion in PB ITO electroes submitte to a large potential step espite the fact that the impeance iagrams plotte by these authors [] for the same electroes in the PB region (which correspons to the initial potential of potential step) clearly show the presence of a large charge transfer resistance compare to the electrolyte an substrate resistances. Pure ohmic rop control was assume by these authors to fit their experimental ata. This shoul be iscusse using a numerical metho to solve the iffusion equations.

22 CHRONOCOULOMETRY The electrical charge variation (potential step chonocoulometry (PSCC) metho) can also be use to characterize insertion processes an etermine the iffusion coefficients an kinetic parameters from experimental ata. The charge is obtaine by integration of the current with respect to time: t Qt ()= I ( τ ) τ (47) an, using imensionless notation, Qt () is ivie by the total amount of charge passe following a potential step, Q= Q( ), an the time ivie by the iffusion time constant τ. 3.. Diffusion controlle processes The Faraaic charge vs. time relationship is erive from Eq. (), for iffusion controlle processes an restricte linear iffusion conitions, as: Q () t 8 = Q π n ( ) n = or, which is equivalent, from Eq. (): π t exp ( n ) 4 τ (48) Q () t t = + Q πτ n= π exp( n τ t) t τ n ( ) n= ( ) n n erfc n t (49) τ The imensionless charge is plotte vs. imensionless time in Fig. 9A. The short-time (Cottrell-omain) expression is obtaine either setting t τ in Eq. (49) or integrating the current with respect to time in Eq. (4): Q() t Q st t = Q t Q t FA Dt () st = Cottrell() = c πτ π as inicate by the ashe lines in Figs. 9B an C using the Q t Q t vs. log t τ an (5) () () ( ) Cottrell Q() t Q vs. t τ graphical representations. Less than % eviation from the limiting Eq. (5) is observe in Eqs. (48) an (49) when t < 3. τ. Uner the above conitions, the iffusion time constant an therefore the iffusion coefficient can be etermine from the slope of the portion of straight line observe for short times in a Qt ()vs. t plot. On the other han, the imensionless charge in Eq. (48) tens towars unity for long times,

23 - 3 - accoring to: Q() t Q lt t = exp 8 π π 4τ [ ] an a log Q () t Q vs. time plot gives a portion of straight line (Fig. 9D) with intercept, log( 8 π ), an slope, π ( 4τ ln ), from which the iffusion coefficient can be obtaine provie t > 4. τ (less than % eviation in Eq. (48) from Eq. (5)). (5) 3.. Influence of insertion reaction kinetics an ohmic rop The influence of insertion reaction kinetics an ohmic rop on the Faraaic charge variation is shown in Fig. using the same imensionless representations as in Fig. 9. The theoretical curve shape epens only on the parameter Λ in Eqs. () an (3), accoring to the following expression erive from Eq. (9) by integration of the current with respect to time: Qt () exp bn t Q = Λ ( τ ) (5) b Λ + Λ+ b n = n ( n) Large Λ values (iffusion control, in the figure) give the same theoretical epenences as in Eq. (48) an Fig. 9. In contrast, small Λ values corresponing to control by interfacial charge transfer kinetics an/or ohmic rop give the following limiting form of Eq. (5): Qt ( ) Q exp Λ tτ (53) Λ = ( ) which can be linearise for short times as: Qt () st, Λ Q= Λ t τ (54) A linear epenence is therefore preicte for the charge vs. time variation in the short-time range (Fig. A) ue to constant current for short times when insertion reaction kinetics an/or ohmic rop control is satisfie. Deviation from iffusion controlle processes an relate Eqs. (48) an (49), ue to limitations by interfacial charge transfer kinetics an/or ohmic rop, is well characterize by the presence of the maximum of the function plotte in Fig. B, which is equivalent to a maximum of the Cottrell function plot for the charge, Qt () tvs. log t, from the experimental point of view. Such a eviation can also be characterize by the inflection point observe in the graphical representation (Cottrell plot for the electrical charge) of Fig. C.

24 - 4 - Finally, from Eq. (5), the long-time variation of Faraaic charge follows the equation: Qt () lt Λ = Q b Λ + Λ+ b ( ) ( ) exp b t τ (55) as illustrate by the semi-logarithmic plot of Fig. D. Whatever the Λ value, a portion of straight line is preicte for long times in a log[ Qt () Q] vs. time plot with slope, b ( τ ln ), an intercept, log { Λ [ b ( Λ + Λ+ b )]}, from which the iffusion coefficient can be obtaine if Λ is not too small. Less than % eviation from Eq. (55) is preicte in Eq. (5) when t > 4. τ for iffusion controlle processes ( Λ ), while a straight line is observe over the whole time omain when Λ, accoring to Eq. (53), ue to rate control by insertion reaction kinetics an/or ohmic rop. As inicate above for the current transient, the theoretical preictions for the Faraaic charge in Eqs. (48) to (55) an Figs. 9 an are vali either consiering small-amplitue potential steps, irrespective of the insertion isotherm type, the reaction rate expression, the presence or not of ohmic rop in the electrolyte, bulk material, etc., an the epenence or not of the iffusion coefficient on the guest species concentration (here the partial erivatives v c an v E in Eq. (3) are calculate at the initial potential E ), or using larger potential steps provie the iffusion coefficient is constant, the Langmuir isotherm satisfie for the insertion process, ohmic rop effects can be neglecte an the partial erivatives v c an v E are evaluate at the applie potential E. 4. DETERMINATION OF THE DIFFUSION COEFFICIENT OF A GUEST SPECIES As inicate in the introuction of this paper, the chemical iffusion coefficients of guest species in host materials are generally etermine either from the limiting Cottrell equation or the exponential ecay of current for short- or long-time experiments, respectively. The error in the etermination of D ue to limitations by surface reaction kinetics an/or ohmic potential rop is iscusse in this Section. Hereafter we assume that the ouble-layer charging current is negligible compare to the Faraaic current over the time range consiere an we use Eqs. (5) an (9) to moel the current transient where the parameter Λ takes on the expressions in Eqs. () an (3). Determination of the iffusion coefficients of guest species by curve fitting is also envisage using the current vs. time an charge vs. time relationships.